NeilWatson wrote:At a certain committee meeting only associate professors and assistant professors are present. Each associate professor has brought 2 pencils and 1 chart to the meeting, while each assistant professor has brought 1 pencil and 2 charts. If a total of 10 pencils and 11 charts have been brought to the meeting, how many people are present?
a) 6
b) 7
c) 8
d) 9
e) 10
Approach 1:
This is a MIXTURE problem.
The MIXTURE:
A total of 10 pencils and 11 charts are brought to the meeting.
Thus, of the 22 items brought to the meeting, the fraction that are pencils = 10/21.
ASSOCIATE professors in the mixture:
Each associate professor brings 2 pencils and 1 chart.
Thus, of every 3 items brought by the associate professors, the fraction that are pencils = 2/3 = 14/21.
ASSISTANT professors in the mixture:
Each assistant professor brings 1 pencil and 2 charts.
Thus, of every 3 items brought by the assistant professors, the fraction that are pencils = 1/3 = 7/21.
To determine the ratio of associate professors to assistant professors in the "mixture", use ALLIGATION.
Step 1: Plot the 3 fractions on a number line, with the fractions for the two types of professors (14/21 and 7/21) on the ends and the fraction for the whole group (10/21) in the middle.
associate 14/21---------------------10/21----------------------7/21 assistant
Step 2: Calculate the distances between the fractions.
associate 14/21---------
4/21--------10/21---------
3/21---------7/21 assistant
Step 3: Determine the ratio in the mixture.
The ratio of associate professors to assistant professors is the RECIPROCAL of the distances in red.
associate : assistant = 3/21 : 4/21 = 3:4.
Since the sum of the parts of the ratio = 3+4 = 7, the total number of professors must be a multiple of 7.
The correct answer is
B.
For another problem that I solved with alligation, check here:
https://www.beatthegmat.com/cars-n-trucks-t115617.html
Approach 2:
Let x = the total number of associate professors and y = the total number of assistant professors.
PENCILS:
Since each associate professor brought 2 pencils, the total number of pencils brought by the associated professors = 2x.
Since each assistant professor brought 1 pencil, the total number of pencils brought by the assistant professors = 1y = y.
Since a total of 10 pencils were brought:
2x+y = 10.
CHARTS:
Since each associate professor brought 1 chart, the total number of charts brought by the associated professors = 1x = x.
Since each assistant professor brought 2 charts, the total number of charts brought by the assistant professors = 2y.
Since a total of 11 charts were brought:
x+2y = 11.
Adding together 2x+y = 10 and x+2y = 11, we get:
(2x+y) + (x+2y) = 10+11
3x + 3y = 21
x+y = 7.
Thus, the total number of professors = 7.
The correct answer is
B.
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